User talk:Gandalf61/Archive5

Mandelbrot set

Instead of just claiming my recent contributions to this article was confusing, could you please explain what exactly you don't understand? This way I can re-write the points I believe are important to this article in a manner that is less confusing. Don't you think the onus is on you to explain your actions since you did the reverting? - Shiftchange 08:52, 18 July 2007 (UTC)

Typically these issues are discussed on the talkpage of the article in question--Cronholm¹⁴⁴ 08:55, 18 July 2007 (UTC)

Shiftchange - happy to explain why I reverted your changes - I have added an explanation to Talk:Mandelbrot set. Gandalf61 09:35, 18 July 2007 (UTC)

As i noted here, the onus is on Shiftchange:

Wikipedia policy is quite clear here: the responsibility for justifying inclusion of any content rests firmly with the editor seeking to include it.

— Wikipedia:Tendentious editing

—Piet Delport 23:22, 18 July 2007 (UTC)

Science, not religion!

Hello Gandalf61! I am RS, a new Wikipedian. I have problem with the article 'Big Bang'. In the section 'Philosophical and religious interpretations', editors of the article have tried to mix Big Bang theory with religion. Big Bang theory is a scientific theory, not a religious or metaphysical theory. Religious views should be removed from the article. Thank you. RS2007 14:31, 19 July 2007 (UTC)

RS2007 - you seem to be expecting me to do something, but I am not sure what. If you want to see what other editors think on this point, you should raise your concerns on the article's talk page, Talk:Big Bang. Gandalf61 14:49, 19 July 2007 (UTC)

AfD/BAJDON 6

Would appreciate hearing your critique on the draft, assuming that this was the actual closing decision and comments. (A.k.a. How was my "closing"?) Thanks! - Best regards, Mailer Diablo 16:36, 22 August 2007 (UTC)

Don't know why you should want my opinion in particular, but FWIW, I think the whole idea of "draft closures" sucks. Gandalf61 19:19, 22 August 2007 (UTC)

your posts

with ref to your posts on http://en.wikipedia.org/wiki/Talk:Bézout's_identity

1. You might notice that all the members of S that you write down are even. Can you see some other properties that all the members of S might have ?

=> doesnt matter from the proofs view point

2. If c is a number that divides both 120 and 36 (a "common divisor"), then it will divide any number of the form 120x+36y - in other words, it will divide every member of S. So 1, 2, 3, 4 , 6 and 12 (the common divisors of 36 and 120) divide every member of S.

=> note we dont know "S" we assume a "d" exists in S. I think just because you are able to factor the numbers here you are able to proceed further. In other words you are working with numbers whose factors can be computed

like I said, just because we already know the factors, we can determine S but however given any generic a and b it may not be possible

In other words, I would not be wrong in saying that your proof is specific to this particular example. —Preceding unsigned comment added by 220.227.207.194 (talk) 12:40, 12 September 2007 (UTC)

I was using an example with small numbers to help you follow it more easily. It is easy to generalise step 2 to any pair of numbers a and b as follows:

2. If c is a number that divides both a and b (a "common divisor"), then it will divide any number of the form ax+by - in other words, it will divide every member of S. So all of the common divisors of a and b divide every member of S.

We don't need to know anything about the factorisation of a and b here. We don't need to know what their common divisors are. And we don't need to know anything about S apart from its definition - it consists of all numbers of the form ax+by. It is an immediate consequence of this definition that any common divisor of a and b will also divide every member of S - because if c divides a and b then it will also divide ax and by and so it will divide the sum ax+by. The other steps in my post can be generalised in the same way. Gandalf61 13:06, 12 September 2007 (UTC)

Fair enough, but why would any of the common divisors of a and b not be a suitable candidate for "d"? —Preceding unsigned comment added by 220.227.207.194 (talk) 13:27, 12 September 2007 (UTC)

Only the largest common divisor of a and b is a candidate for d (the smallest member of S) because none of the smaller common divisors can be in S at all - because we know from step 2 that every member of S must be a multiple of the largest common divisor of a and b, but the smaller common divisors are not multiples of the largest (because they are smaller than it). This is step 4 in my post. Gandalf61 13:42, 12 September 2007 (UTC)

ok, ill step out how i got to the d=1 case

let c1,c2,...cn be the set of common divisors of a and b Let C=c1.c2.c3...cn = gcd(a,b) a=C.a1 ; b=C.b1; a1 and b1 are coprime to each other..agree? so C.a1x+C.b1y=d ? a1x+b1y=d/C Hence we have integers on the left and a fraction on the right so d must be divisible by C say d=kC so we now have to show that there exists k such that a1x+b1y=k , where k is any integer >0 ? now how do I prove k=1? ; given a1 and b1 are coprime to each other?

I may be missing something but the fact that C =gcd(a,b) can exist is only valid if the a1x+b1y=1 exists, is that correct or not?

So the proof still seems to assume it is correct before it starts off, agree? Please dont use the remainder theorem to prove k=1 again, because k=1 is true for all values of a and b using the division algorithm; in other words if I use the division algorithm, I can always show r=0 even if a and b are not coprime. So the proof does assume that say if "c" does not exit, the remainder theorem is sufficent to prove ax+by=d exists? but d=1 breaks it.

Sorry if I am trying your patience, but am a bit grey

—Preceding unsigned comment added by 220.227.207.194 (talk) 07:32, 13 September 2007 (UTC) 

Just to add, If you are going with the assumption that given ax+by=1 is always true if a and b are coprime (for which I still have to see a formidable proof, dont say division algorithm because the division algorithm says ax+by=1 is possible even if a and b are not coprime); then there is nothing left to prove, because it simply means the rest is extrapolated by multiplying both sides by a common factor, which is what the identity states. However I still havent seen a formidable proof for ax+by=1 being true for all cases where a and b are coprime —Preceding unsigned comment added by 220.227.207.194 (talk) 09:28, 13 September 2007 (UTC)

Three easy steps:

1. Out of all the common divisors of a and b, the only candidate that can possibly be in S is the greatest common denominator, gcd(a,b), which we are now calling C.

=>ok. So you are saying "C" is a likely candidate.

• Read carefully what I am saying. If there is a common divisor of a and b in S then it can only be their greatest common divisor, which we are calling C (notice this is a capital C - this is the notation that you introduced above). I have not said anything about whether this is likely or not likely.

2. The smallest positive member of S, which we call d, is a common divisor of a and b - because if it was not a common divisor of a and b then we could find a positive member of S that is smaller than d, which contradicts the definition of d. This is the proof outlined in the article - it is a proof by contradiction.

=>ok, if there was no "C" i.e given a and b were coprime then?

Does it not mean you have already assumed that for 2 coprime numbers a and b, ax+by=1 is true?

• No. C is the greatest common divisor of a and b. The greatest common divisor always exists. It might be 1 if a and b are coprime - but I am not assuming that it is 1. We know that C exists, but I am not assuming anything else about it.

3. Therefore, from 1 and 2, d (the smallest positive member of S) must equal C (the greatest common denominator of a and b).

The proof of Step 2 uses the division algorithm (is this what you are calling the "remainder theorem" ?).

=> yes am correcting it above

But it does not assume that if a and b are coprime then we can find x andy such that ax+by=1. This fact is a consequence of Bezout's identity - it is the special case when gcd(a,b)=1.

=>exactly..but isnt that what you have already assumed if there was no such "c" in the 1st place? example: you have assumed there is a set S , d is the smallest element, by the division algorithm you have shown r=0, if d=1 which means you have already assumed that ax+by=1 is a valid combination if no "C" exists, correct? incorrect?

• Incorrect. Nowhere in the proof is it assumed that d=1. Nowhere. Clear this out of your mind. You are imagining it. And don't mix up c with C. Remember we are using C for gcd(a,b), which always exists.

So we cannot use this fact in the proof of Bezout's identity - that would be circular reasoning. Read the outline proof given in the Bezout's identity article again - it does not assume this fact.

=> the proof says "if there is a c such that.." now assume we know there is no such "C" ; simply said no such "C" exists, so we have

r=a(1-qx)-bqy

Now if a and b are co prime, what can we say about d? a=qd+r, so we have to assume that d=1 is a possible candidate, is it not? if we do not make the assumption we cannot show r=0. and if we are making that assumption we are assuming that the identity holds anyways for coprimes a and b.

so if we assume d=1, to show r=0 we have

r= a(1-ax)-bay=a(1-ax-by)? right? now if want to have r=0, have we not already assumed 1-ax-by=0 ? because to get here we have already assumed d=1 is true?

So in other words the moment you have assumed d=1 is a likely candidate you have already assumed 2 coprime numbers a and b can exist so that ax+by=1

If you can explain this specific part above, (given no "c" exists) it would be great.

• To get from r=a(1-qx)-bqy to r=a(1-ax)-bay you have to assume that a=q, which can only be true if d=1 and r=0. Notice that you are assuming this - not the proof. The proof does not say this. It does not say that r=a(1-ax)-bay. It just says that because r=a(1-qx)-bqy then r is in S. And we know (from the division algorithm) that 0<=r<d. And we know that d is the smallest positive member of S because we have defined it that way. So the only way to satisfy all of these statements is if r is 0. The proof does not make any claims about the value of d. It does not assume that d=1. We don't know the value of d. We don't need to know the value of d.

• Read the proof carefully. Read it slowly. Stop bringing you own assumptions with you into the proof, and it might become clearer to you. The assumption that d=1 is an assumption that you are making in the first place. It is not in the proof. Gandalf61 13:43, 13 September 2007 (UTC)

Quoting the proof:

If c is another common divisor of a and b, then c also divides ax + by = d.

Now assume no such c exists. There is no common divisor between a and b but "d" alone. We do not enter the if condition at all. In other words there exist a d element of S How do you proceed with the proof?

we have shown that by the division algorithm, 0<=r<d is necessary else r is an element of S and r<d

So you are saying the proof says: The only way to satisfy all of these statements is if r is 0. So if r is 0, as the proof assumes/says is the only way to satisfy this, let us see the effect: so a=qd is true? likewise b=sd+r1 now b=sd also has to be true? and the same logic for r1? r1 has to be 0?

now we know "no other c exists which divides both a and b", we have a=qd , b=sd now if a=qd and b=sd , and no other "c" exists which divides both a and b, does it not imply that there is only one number d which is common divisior of both a and b? if for example another number c existed then both c and d would divide a and b? now 1 is the only number which can always divide any 2 integers a and b, is that not true? if there existed another value for d but 1, that means there are 2 common factors between a and b, but we have already said there is no other common factor "c" but "d" alone. so at this point itself you have implied that this number d has to be 1? is my reasoning correct? Note this is a direct consequence the assumption that r=0 and r1=0

Is the fact that r=0 and r1=0 not directly leading to this? In other words the fact that the proof "assumes" or "says r=0" is necessary, it has indirectly said that if "no c exists, d=1" so perhaps the assumption that r=0 is misleading r>d could be an outcome too for example. But the moment you have said/stated/assumed r=0, and no other c exists you have stated that any 2 coprime integer numbers a and b satisfy ax+by=1 , where x and y are positive. So perhaps r=0 is an assumption that needs to be checked, it has been taken too trivially. —Preceding unsigned comment added by 220.227.207.194 (talk) 14:15, 13 September 2007 (UTC)

The proof proves that r=0 and r1=0 - so d, the smallest positive member of S, is a divisor of a and a divisor of b. It does not assume this - it proves it using the division algorithm. We know that r cannot be greater than d because the division algorithm says that a=qd+r and 0<=r<d. r is the remainder left when you divide a by d. If r was greater than d then you could just increase q to make r smaller.

Now, let's start from the point where we have proved that d, the smallest positive member of S, is a common divisor of a and b. The proof in the article says that if there is some other common divisor c of a and b then c must divide d because d=ax+by for some integers x and y. So if c, this other common divisor, exists then it must divide d, and so d is the greatest common divisor.

But you are asking what happens if c, this other common divisor, does not exist. In this case d is the only common divisor of a and b. So d is certainly the greatest common divisor because it is the only common divisor. And, incidentally, in this case d will be 1 because 1 is always a common divisor of any two numbers.

So d is still the greatest common divisor, no matter whether some other common divisor c exists or not. The proof does not assume that c exists. It does not need to assume that. It does not assume anything about c at all. Gandalf61 15:08, 13 September 2007 (UTC)

another dimension to the above

hmm what if d>a ? a=qd+r , 0<=r<|d| , can you show a suitable choice for d? By bringing in the division algorithm , which I have been pouring over last night,i was specifically thinking of the following case, say 7=9q+r , can I find such an r which satisfies the division algorithm? or do we make an assumption a>=d and b>=d look if we make the assumption that a>=d and b>=d then the proof for Bezout's identity is no rocket science: ax+by=d , if d has to have a common factor with a and b? if a and b are coprime d still has to have a common factor with a and b? (simple if a and b are not coprime let g be the common factor, then ga1x+gb1y=d implies g has to divide d, else we have integers on the left and fraction on the right, so d could be gk where k is any integer a1x+b1y=k) so the proof still reduces to: given 2 coprimes a and b, ax+by=d ,

now we know d has to have a common factor with a and b; (note I said common factor, i did not say divisible by both a and b because there are values like 2x+5y=7 , where 7 is not divisible by either 2 or 5)

if we "assume" d<=a and d<=b (is this assumption valid because by bringing in the division theorem you have already made this assumption) then there is no need for the division algorithm, we just need to assume d<a and d<b and 1 is the right candidate.

So is making the assumption d<=a and d<=b right? Note the Division algorithm does seem to imply this. So if we simply assumed d<=a and d<=b we would be able to prove the identity without the algorithm itself. In other words, we need a number which has a common fact with a and b, it has to be <=a and <=b ; so 1 is always the best choice. and will always fit in for any coprime a and b

The division algorithm still works if d>a - just set q equal to 0 and r equal to a. But in our case we know that d<=min(a,b) anyway, because both a and b are in S and d is the smallest positive member of S. We don't have to assume that d<=min(a,b) - it follows immediately from the definition of d. Gandalf61 13:47, 14 September 2007 (UTC)

bummer? does the divison algorithm assume d<=a, yes or no?\ setting q=0 is an assumption the proof does not say so, "you" are assuming it :)

The division algorithm is true for any two integers a and d as long as d≠0. No other assumptions are made in the division algorithm about the relative magnitudes of a and d. If 0≤a<|d| then we can set r=a (which satisfies the constraint 0≤r<|d|), and this makes q equal to 0, because a=0.d + a. You raised the question "what if d>a", and there are no assumptions in here at all - it's all a simple consequence of the division algorithm. This is really very obvious, and if you don't follow it, then you need to read the division algorithm article really carefully. Gandalf61 16:06, 14 September 2007 (UTC)

was looking up the uniquness condition for the division algo if q=0, anyways the case that d<=a and d<=b does not need the division algo at all (whee hoo ! new proof for the identity!)

r=a(1-qx)-bqy , if we assume d>a , q=0, r=a and there isnt much we can say as this is not of the form r=aX+bY, this is of the form X=1,Y=0 in r=aX+bY. So we cannot proceed further as i see it, however as i see it d<=a d<=b , if assumed does not need the division algorithm , and if we do not assume it we are stuck with r=a

'Coast of Britain' Article

i really enjoy the "how long is the coast of britain?" article. Thanks for adding and maintaining it! —Preceding unsigned comment added by 65.216.75.240 (talk) 20:18, 18 September 2007 (UTC)

Glad to hear you enjoyed the article. Of course, many other editors have worked on it as well, so I can't take more than a small part of the credit. Gandalf61 06:29, 19 September 2007 (UTC)

Thank you

Thanks for your good wishes. I'm not enjoying editing right now, so your kind words were very much appreciated. --Dweller 10:49, 1 October 2007 (UTC)

You're welcome. RD-TWA is one of the highlights of my week - keep it up. Gandalf61 12:27, 1 October 2007 (UTC)

silicon, bismuth and water-referring to your user page

I believe silicon also is denser as a liquid than as a solid. -Rich Peterson —Preceding unsigned comment added by 130.86.14.84 (talk) 03:38, 3 October 2007 (UTC)

Interesting. That's yet another thing I have learned from Wikipedia. Gandalf61 08:19, 3 October 2007 (UTC)

Ref Desk

No problem. I've noticed Clem's civility problems before, though, so I doubt it'll make much impact. — Lomn 14:31, 1 November 2007 (UTC)

List of Snowclones

I hadn't realized the list of snowclones was gone. I think you mentioned archiving it somewhere, but I didn't see it at first glance. Is there a simple way to see the old article, either from a copy or otherwise?

On a vaguely similar note, if someone is going to make a list of erdos numbers, wouldn't that be amazingly easier while the "what links here" button still works? —Preceding unsigned comment added by JackSchmidt (talk • contribs) 16:13, 2 November 2007 (UTC)

No, sorry, I don't have an archive copy of the list of snowclones. I did ask the deleting admin, User:Neil, if he would place a copy of the article in my user space (see here) - I think admins can do this, even after an article has been deleted. But he didn't respond, and I haven't chased him about it either. Gandalf61 16:38, 2 November 2007 (UTC)

Thanks, I asked under the deletion review topic on his talk page. Seems tricky to handle deletes. One needs to be careful to really eradicate copyright violations, libel, and such, but if one was going to keep the snowclones and the erdos numbers, where would you put them? At any rate, I said I'd be happy if the snowclones were in your user area. JackSchmidt 17:06, 2 November 2007 (UTC)

Hi Gandalf61 - I missed Jack's first request, because my talk page has been mental. It's been userfied to User:JackSchmidt/List of snowclones (he asked me first!). As I said to Jack on his talk page, please don't move it to article space or copy thew entire thing unreferenced to the snowclones article. Neil ☎ 17:41, 4 November 2007 (UTC)

Categories by Erdos numbers

User:Mikkalai/By Erdos contains a very raw list made from remnants of categories and the log of the bot which implemented the deletion you opposed. Please join the discusion here to decide how to proceded. A clandestinely proud Erdos-Number-3-wikipedian `'Míkka 16:27, 2 November 2007 (UTC)

Deletion of the Erdos Number categories

Recently, as you know, the categories related to Erdos Number were deleted. There are discussions and debates across several article talk pages (e.g. the Mathematics WikiProject Talk page. I've formally requested a deletion review towards overturning the deletion, at this deletion review log item. Pete St.John 21:07, 7 November 2007 (UTC)

Lucas sequence

${\sqrt {x}}$ means a unique number, not two values, so the expressions are not symmetric in a,b.--Patrick (talk) 11:58, 29 December 2007 (UTC)

Don't forget that $P^{2}-4Q$ may be negative - the only case that is excluded is $P^{2}-4Q=0$, in which case the characteristic equation would have a double root. If $P^{2}-4Q$ is negative then ${\sqrt {P^{2}-4Q}}$ is imaginary, and may represent either of the square roots of $P^{2}-4Q$. Note that ${\frac {a^{n}-b^{n}}{a-b}}$ is symmetric in a,b (interchanging a and b does not alter the value of the expression), and it is fairly clear in the alternative expression ${\frac {a^{n}-b^{n}}{\sqrt {P^{2}-4Q}}}$ that ${\sqrt {P^{2}-4Q}}$ is to be taken in the sense makes it equal to $a-b$. However, if you think that the expression ${\frac {a^{n}-b^{n}}{\sqrt {P^{2}-4Q}}}$ is ambiguous then you could remove it - it is not essential. Gandalf61 (talk) 12:31, 29 December 2007 (UTC)

There are two more formulas with the square root. I fixed it without throwing away all these.--Patrick (talk) 14:30, 29 December 2007 (UTC)

Well, I think you've introduced a rather ugly asymmetry into that section, which was cleaner and clearer as it was before - but it's not a big issue. Gandalf61 (talk) 15:25, 29 December 2007 (UTC)

It was wrong. In some contexts a square root might be multi-valued, but that would have to be made explicit, and would require additional explanation what you mean by an equality etc. Normally a square root has one value. Also, having different expressions for $a^{n}$ and $b^{n}$ is not compatible with symmetry.--Patrick (talk) 23:15, 29 December 2007 (UTC)